| L(s) = 1 | − 0.449·3-s + 0.787·5-s − 2.79·9-s + 3.10·11-s − 5.59·13-s − 0.354·15-s − 5.38·17-s − 4.82·19-s + 7.02·23-s − 4.37·25-s + 2.60·27-s − 7.52·29-s − 4.90·31-s − 1.39·33-s − 9.21·37-s + 2.51·39-s + 41-s + 11.7·43-s − 2.20·45-s + 5.56·47-s + 2.41·51-s + 5.94·53-s + 2.44·55-s + 2.16·57-s + 10.2·59-s − 4.89·61-s − 4.40·65-s + ⋯ |
| L(s) = 1 | − 0.259·3-s + 0.352·5-s − 0.932·9-s + 0.935·11-s − 1.55·13-s − 0.0914·15-s − 1.30·17-s − 1.10·19-s + 1.46·23-s − 0.875·25-s + 0.501·27-s − 1.39·29-s − 0.881·31-s − 0.242·33-s − 1.51·37-s + 0.402·39-s + 0.156·41-s + 1.79·43-s − 0.328·45-s + 0.811·47-s + 0.338·51-s + 0.816·53-s + 0.329·55-s + 0.287·57-s + 1.32·59-s − 0.626·61-s − 0.546·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.086621308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.086621308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 0.449T + 3T^{2} \) |
| 5 | \( 1 - 0.787T + 5T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 7.02T + 23T^{2} \) |
| 29 | \( 1 + 7.52T + 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 - 8.10T + 79T^{2} \) |
| 83 | \( 1 + 2.10T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 2.23T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69941605432097180984537465579, −7.02690394742191988589982814260, −6.53181700349933908426536802824, −5.66328390052696794503315664201, −5.17457485276175355378250680396, −4.30827888408815335976820084695, −3.58816354713439329243140517190, −2.40471037719983938082199794021, −2.03359780429162876812007776166, −0.49195959239727613417242393195,
0.49195959239727613417242393195, 2.03359780429162876812007776166, 2.40471037719983938082199794021, 3.58816354713439329243140517190, 4.30827888408815335976820084695, 5.17457485276175355378250680396, 5.66328390052696794503315664201, 6.53181700349933908426536802824, 7.02690394742191988589982814260, 7.69941605432097180984537465579
